Question: If $n=2^3 \cdot 3^2 \cdot 5$, how many even positive factors does $n$ have?
Explanation: A positive integer is a factor of $n$ if and only if its prime factorization is of the form $2^a\cdot 3^b\cdot 5^c$ where $0\leq a\leq 3$, $0\leq b\leq 2$, and $0\leq c\leq 1$.  An integer is even if and only if the exponent of 2 in its prime factorization is at least 1.  Therefore, we have 3 choices for $a$, 3 choices for $b$ and $2$ choices for $c$, for a total of $(3)(3)(2)=\boxed{18}$ ways to form an even positive factor of $n$.